# On Î²-Open Sets and Ideals in Topological Spaces

## DOI:

https://doi.org/10.29020/nybg.ejpam.v12i3.3438## Keywords:

$\beta$-open sets, $\beta_{I}$-open sets, $\beta_{I}$-compactness, $c\beta_{I}$-compactness, $\beta_{I}$-hyperconnectedness and $c\beta_{I}$-hyperconnectednes## Abstract

Let X be a topological space and I be an ideal in X. A subset A of a topological space X is called a Î²-open set if A âŠ† cl(int(cl(A))). A subset A of X is called Î²-open with respect to the ideal I, or Î²I -open, if there exists an open set U such that (1) U âˆ’ A âˆˆ I, and (2) A âˆ’ cl(int(cl(U))) âˆˆ I. A space X is said to be a Î²I -compact space if it is Î²I -compact as a subset. An ideal topological space (X, Ï„, I) is said to be a cÎ²I -compact space if it is cÎ²I -compact as a subset. An ideal topological space (X, Ï„, I) is said to be a countably Î²I -compact space if X is countably Î²I -compact as a subset. Two sets A and B in an ideal topological space (X, Ï„, I) is said to be Î²I -separated if clÎ²I (A) âˆ© B = âˆ… = A âˆ© clÎ²(B). A subset A of an ideal topological space (X, Ï„, I) is said to be Î²I -connected if it cannot be expressed as a union of two Î²I -separated sets. An ideal topological space (X, Ï„, I) is said to be Î²I -connected if X Î²I -connected as a subset. In this study, we introduced the notions Î²I -open set, Î²I -compact, cÎ²I -compact, Î²I -hyperconnected, cÎ²I -hyperconnected, Î²I -connected and Î²I -separated. Moreover, we investigated the concept Î²-open set by determining some of its properties relative to the above-mentioned notions.

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*European Journal of Pure and Applied Mathematics*,

*12*(3), 893–905. https://doi.org/10.29020/nybg.ejpam.v12i3.3438

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